Valuation of Convertible Bonds pdf

Sanaa Khan K1306336

FACULTY OF SCIENCE, ENGINEERING AND COMPUTING

School of Computer Science and

Mathematics

BSc (Hons) DEGREE

IN

Financial Mathematics with Business Management

Name: Sanaa Khan

ID Number: K1306336

Project Title: Valuation of Convertible Bonds

Date: 11/04/16

Supervisor: Luluwah Al-Fagih

WARRANTY STATEMENT

This is a student project. Therefore, neither the student nor Kingston University

makes any warranty, express or implied, as to the accuracy of the data or conclusion

of the work performed in the project and will not be held responsible for any

consequences arising out of any inaccuracies or omissions therein.

Abstract

In this paper, we will be discussing methods of pricing a European style convertible bond

(CB), i.e. where conversion can only take place at maturity. Pricing methods include using

the Black-Scholes model to price the bond by splitting components to help simplify the

procedure. Furthermore, contract features will be looked upon, to give a better perspective

as to what is said between the issuer and the bondholder, as well as how the CB is formed

and the features within it. The paper will also be looking at the analysis of price sensitivities

and how different features affect the price of a CB and the impact they have on a portfolio

containing a CB.

Sanaa Khan K1306336

Contents

Abstract i

1. Introduction 1

2. Payoff Profiles 4

2.1. Notation 4

2.2. Bondholders Perspective 5

2.3. Bond Issuers Perspective 5

2.4. Payoff 6

2.5. A Zero-Sum Game 7

2.5.1. Example 7

3. Contract Features of a Convertible Bond 9

3.1. Convertible Bond Financing 9

3.2. Maturity 9

3.3. Principle 9

3.4. Conversion Ratio 10

3.5. Call Provisions 10

3.6. Put Provisions 10

3.7. Coupon Payments 11

3.8. Refix Clause 11

3.9. Other Non-Standard Clauses 12

3.10. Termination 12

Sanaa Khan K1306336

4. Properties of a Convertible Bond 13

4.1. Conversion Price 13

4.2. Parity 13

4.3. Premium to Parity 14

4.4. Investment Premium 14

4.5. Bond Floor 15

4.6. Price Sensitivities 16

4.7. Upper and Lower Bounds 19

5. Pricing Methods 21

5.1. Mathematical Background 21

5.2. Monte Carlo Simulation 21

5.3. Lattice based Method 22

5.4. Reduced Form Approach 22

5.5. Tsiveriotis-Fernandes Method 23

5.6. Black-Scholes Method 23

6. Black-Scholes Model 26

6.1. The Bond Price 26

6.2. Example 29

6.3. Margrabe Formula 31

7. Conclusion 34

8. References 35

1

1. Introduction Convertible bonds (CB) were first used during the 1960s. Convertible bonds are hybrid

securities; they use both equity and debt. A convertible bond is a bond such that the holder

of the bond; that being the investor is able to convert it into cash or equity when they feel it

would be beneficial to them [7 - pg 58]. Ingersolls (1977) research suggests that the general

valuation procedure would be to set up the price of the convertible and equate it to the

maximum value of a straight bond, or the value it holds within the common stock (after

conversion) given that at some point in the near future. The value found from this, would

then be discounted back to the present value. Yan, Yi, Yang and Liang (2015) state they wish

to keep hold of the bond, in which case they will receive interest payments; or they could

convert it into the companys stocks. The bondholder would ideally pick a strategy in which

they would be able to maximise the CB value.

The issuers of convertible bonds are usually smaller firms. Smaller firms who are looking into

getting finances. The reason for this is because smaller firms are not as well-known and need

financing when their credit is low [20]. It is found that when a weaker firm wishes to issue a

CB, it shows they have faith in their project. This enhances their chances of gaining investors

for their company. However, a larger firm would not need to issue convertible bonds as they

would easily be able to get funding and or loans as they are more known within the industry.

If a larger firm wanted to issue a bond, they would not have enough buyers.

The motivation behind the smaller firms issuing the convertible bonds is due to the fact they

lack stable credit histories. This means they would have to pay higher interest payments;

also known as coupons - to their debt holders. The size of a firm usually is a reason as to why

there is an issuance of convertible bonds [12]. Firm size is associated with bankruptcy costs;

since smaller firms are more vulnerable to failure and are risk averse. Smaller firms face

Sanaa Khan K1306336

2

higher degree information asymmetry, this could increase the cost of the debt. It could also

lead to having more restrictive contracts also known as covenants, had they wanted to

issue a straight bond. This is a reason why larger firms just offer straight bonds. A convertible

bond is more flexible the way it works, matters are stated within the contract, as well as

being set out if the firm breaks the contract they (the bondholder) will receive a premium

[12]. The motivation behind issuing CBs is the fact firms will have interest rate-cost savings,

in comparison to issuing straight corporate1 bonds [20].

Another reason why firms issue CBs is to ensure the investor has no entitlement in the

running of the business. This would mean having the ability to vote for the directors that

would only be in control of the common stockholders. This makes it attractive to firms, as

they know their positions will not be endangered nor questioned. Kwok (2014) suggests that

convertible bonds are chosen by firms over straight bonds due to the lower coupon rate.

CBs have a callable feature which means it can be redeemed by the issuer prior to the

contractual date, this paper will follow a European styled CB. At this point, a price in the

form of a penalty, would be paid to the bondholder, as the company is forcing them to

either convert or surrender the bond [ref 7 page 58].

Owning a convertible bond is like playing a game. The bondholder is allowed to convert the

bond when they see it is beneficial for them. Suppose the bondholder converts before the

call date set within the covenant; it would mean the shareholders were not able to call the

bond when they thought it would be beneficial for them [7]. According to Yan, Yi, Yang and

Liang (2015) when the coupon rate is bounded above by the interest rate multiplied by the

strike price, that is when the bondholder will convert the CB. The conversion for the issuer

will take place when the coupon rate is lower than the dividend rate multiplied by the strike

1 Information Asymmetry: a party within a transaction has more information on the other party that

they are dealing with. Due to this, a party is likely to take advantage of the other partys lack of knowledge.

Sanaa Khan K1306336

3

price; though this paper will not be discussing dividends used within CBs. The contract is

terminated when the coupon rate lies in between the two bounds, at that point both parties

will terminate the contract.

The bondholder will be receiving coupon payments, over the life of the CB, up until the

contract has reached its expiry (maturity). Prior to maturity, the bondholder has the right to

convert their bond into the companys shares. Close to the end of the contract the company

have the right to call the bond back and force the bondholder to capitulate the bond to the

company.

In this paper, we will be discussing methods of pricing a European style convertible bond

(CB), i.e. where conversion can only take place at maturity. Pricing methods include using

the Black-Scholes model to price the bond by splitting components to help simplify the

procedure. Furthermore, contract features will be looked upon, to give a better perspective

as to what is said between the issuer and the bondholder, as well as how the CB is formed

and the features within it. The paper will also be looking at the analysis of price sensitivities

and how different features affect the price of a CB and the impact they have on a portfolio

containing a CB.

Sanaa Khan K1306336

4

2. Payoff Profiles

A payoff is what is received by the bondholder during the lifetime of the bond. The

bondholder has two options, (i) to receive the face value, or (ii) the share price multiplied by

the conversion ratio the one with the greater value is what the bondholder will receive.

First we introduce some notation that will be used throughout the paper.

2.1 Notation

Face value

Conversion ratio

Share price at time

Coupon payment

Maturity

Conversion price

Strike price

Bond floor